L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 2·11-s + 12-s − 14-s + 3·15-s + 16-s + 7·17-s − 18-s + 6·19-s + 3·20-s + 21-s − 2·22-s − 24-s + 4·25-s + 27-s + 28-s − 3·30-s + 6·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.603·11-s + 0.288·12-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s + 1.37·19-s + 0.670·20-s + 0.218·21-s − 0.426·22-s − 0.204·24-s + 4/5·25-s + 0.192·27-s + 0.188·28-s − 0.547·30-s + 1.07·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.960561448\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.960561448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.50318217849137, −14.88391501688775, −14.39004275929534, −13.95259057842778, −13.57787810278974, −12.85979732239676, −12.12875514906790, −11.74516702384021, −11.06784504129507, −10.12958732120974, −9.980492368322124, −9.547359266780906, −8.929694137263270, −8.327028743899767, −7.737177353378321, −7.189908822044899, −6.501642790431075, −5.772704320376416, −5.407283940206991, −4.542008937911672, −3.535888086217377, −2.948164028555377, −2.226382511234245, −1.367445065934231, −1.024307962661158,
1.024307962661158, 1.367445065934231, 2.226382511234245, 2.948164028555377, 3.535888086217377, 4.542008937911672, 5.407283940206991, 5.772704320376416, 6.501642790431075, 7.189908822044899, 7.737177353378321, 8.327028743899767, 8.929694137263270, 9.547359266780906, 9.980492368322124, 10.12958732120974, 11.06784504129507, 11.74516702384021, 12.12875514906790, 12.85979732239676, 13.57787810278974, 13.95259057842778, 14.39004275929534, 14.88391501688775, 15.50318217849137